Yet More CP2-7 Hands

[MarkGritter]

[ QUOTE ]

I was surprised to see the *boats* at 64% instead of 74%, is that right?

[/ QUOTE ]

Err, sorry. Completely missed what you where getting at.

The right number is really 64%. Sixes full would normally only be 70th percentile in back, but when we take the dead cards (from our hand) into account our opponent will have a better hand 36% of the time:

<font class="small">Code:</font><hr /><pre>
full house Fives full , 2077 , 0.642789
full house Nines full , 7666 , 0.703015
full house Jacks full , 9208 , 0.775356
full house Queens full , 9333 , 0.848679
full house Kings full , 9034 , 0.919653
full house Aces full , 2869 , 0.942193
quad Deuces , 1073 , 0.950623
quad Threes , 1059 , 0.958943
quad Nines , 998 , 0.966783
quad Jacks , 981 , 0.974491
quad Queens , 1012 , 0.982441
quad Kings , 975 , 0.990101
straight flush Six-high , 217 , 0.991806
straight flush Jack-high , 198 , 0.993361
straight flush Queen-high , 174 , 0.994728
straight flush King-high , 399 , 0.997863
straight flush Ace-high , 272 , 1.000000
</pre><hr />

[Sweet]

[ QUOTE ]
[ QUOTE ]

I was surprised to see the *boats* at 64% instead of 74%, is that right?

[/ QUOTE ]

Err, sorry. Completely missed what you where getting at.

The right number is really 64%. Sixes full would normally only be 70th percentile in back, but when we take the dead cards (from our hand) into account our opponent will have a better hand 36% of the time:

<font class="small">Code:</font><hr /><pre>
full house Fives full , 2077 , 0.642789
full house Nines full , 7666 , 0.703015
full house Jacks full , 9208 , 0.775356
full house Queens full , 9333 , 0.848679
full house Kings full , 9034 , 0.919653
full house Aces full , 2869 , 0.942193
quad Deuces , 1073 , 0.950623
quad Threes , 1059 , 0.958943
quad Nines , 998 , 0.966783
quad Jacks , 981 , 0.974491
quad Queens , 1012 , 0.982441
quad Kings , 975 , 0.990101
straight flush Six-high , 217 , 0.991806
straight flush Jack-high , 198 , 0.993361
straight flush Queen-high , 174 , 0.994728
straight flush King-high , 399 , 0.997863
straight flush Ace-high , 272 , 1.000000
</pre><hr />

[/ QUOTE ]

I find this short table very philosophically significant. I am shocked to see 9s full at 70% as opposed to its usual 80%. It sort of quantifies how drastically the hand values change based on the cards you have in your hand. If realizing there is a shift in values is step 1, then certainly quantifying it is step 2. I would love to see more little tables like this one, where we can see the skew in one (or more) of the expected distributions of the opponents three hands based on the cards we hold. Any more interesting examples? Extreme examples? Examples with almost no skew? I always feel like have a long flush or a short suit makes it more likely for my opponent to have a flush in back... how right or wrong am I? How much does the front skew when we have no aces?

Thanks!

[MrMemory]

Gritter for president.

[Sweet]

Is there a hand that I can have that shifts the percentile of some possible front, middle, or back hand of my opponent by more than 20%? More than 15%?

[MarkGritter]

Yes. Here are a couple examples:

Distribution of original 10M-hand middles:
0.050031, pair of Deuces KQT kickers
0.100094, A9876
0.150148, KT765
0.200120, QT976
0.250363, JT984
0.300508, J9852
0.351011, J6542
0.400702, T9642
0.451190, T7654
0.503044, 98753
0.550943, 98542
0.603604, 97542
0.655556, 95432
0.706090, 87632
0.757198, 87432
0.813030, 86532
0.857918, 85432
0.916812, 76532
0.951128, 76432

Distribution of middles after removing 2c2s2h3c3s3h4c4s4h5c5s5h7d:

0.050008, pair of Tens J76 kickers
0.100024, pair of Sevens 965 kickers
0.150118, AKT75
0.200172, AJ987
0.250353, KQT98
0.300345, KJ987
0.350071, K9765
0.400141, QJ954
0.450219, QT873
0.500447, Q8654
0.550603, JT872
0.600439, J9863
0.650188, J7654
0.700188, T9764
0.750282, T8754
0.805085, 98763
0.852758, 98642
0.900799, 96532
0.952053, 87542

This is a more than 20% shift in 87542. (Incidentally, the best setting for those cards is 55544 22247 333, which loses 0.87 points on average. 34455 23457 223 loses 1.63 points instead.)

Second example, original front distribution:

0.051687, QJT
0.106333, KQJ
0.151715, AQ3
0.200677, AK8
0.252720, AKQ
0.300212, pair of Eights Six kicker
0.350101, pair of Tens Four kicker
0.400430, pair of Jacks Seven kicker
0.452320, pair of Queens Three kicker
0.509109, pair of Queens King kicker
0.552476, pair of Kings Five kicker
0.601870, pair of Kings Jack kicker
0.676056, pair of Kings Ace kicker
0.706992, pair of Aces Six kicker
0.772863, pair of Aces Jack kicker
0.808908, pair of Aces Queen kicker
0.870549, pair of Aces King kicker
0.912963, trip Fours
0.951167, trip Sevens

Distribution after removing AAAKKKQQJJTT9
0.051001, T82
0.100480, J98
0.151278, Q92
0.200873, QJ6
0.251219, KJ4
0.300369, A65
0.351558, AQ6
0.400184, pair of Deuces Five kicker
0.451209, pair of Fives Queen kicker
0.502008, pair of Sevens Nine kicker
0.555665, pair of Eights Queen kicker
0.601940, pair of Nines Queen kicker
0.651645, pair of Jacks Five kicker
0.701717, pair of Queens Five kicker
0.793120, trip Deuces
0.835685, trip Threes
0.875156, trip Fours
0.909324, trip Fives
0.966090, trip Sevens

QQ6 moves from &lt; 50th percentile to 70.